Replace magic constant test for linear solve on triangular matrices

NEWS.md

@@ -115,6 +115,8 @@ Library improvements
 
       * Mutating linear algebra functions no longer promote ([#5526]).
 
+      * `condskeel` for Skeel condition numbers ([#5726]).
+
     * Sparse linear algebra
 
       * Faster sparse `kron` ([#4958]).
@@ -233,6 +235,7 @@ Deprecated or removed
 [#5475]: https://github.com/JuliaLang/julia/pull/5475
 [#5526]: https://github.com/JuliaLang/julia/pull/5526
 [#5538]: https://github.com/JuliaLang/julia/pull/5538
+[#5726]: https://github.com/JuliaLang/julia/pull/5726
 
 Julia v0.2.0 Release Notes
 ==========================

base/exports.jl

@@ -590,6 +590,7 @@ export
     cholpfact!,
     cholpfact,
     cond,
+    condskeel,
     cross,
     ctranspose,
     det,

base/linalg.jl

@@ -45,6 +45,7 @@ export
     cholpfact,
     cholpfact!,
     cond,
+    condskeel,
     copy!,
     cross,
     ctranspose,

base/linalg/generic.jl

@@ -174,6 +174,12 @@ end
 cond(x::Number) = x == 0 ? Inf : 1.0
 cond(x::Number, p) = cond(x)
 
+#Skeel condition numbers
+condskeel(A::AbstractMatrix, p::Real=Inf) = norm(abs(inv(A))*abs(A), p)
+condskeel{T<:Integer}(A::AbstractMatrix{T}, p::Real=Inf) = norm(abs(inv(float(A)))*abs(A), p)
+condskeel(A::AbstractMatrix, x::AbstractVector, p::Real=Inf) = norm(abs(inv(A))*abs(A)*abs(x), p)
+condskeel{T<:Integer}(A::AbstractMatrix{T}, x::AbstractVector, p::Real=Inf) = norm(abs(inv(float(A)))*abs(A)*abs(x), p)
+
 function issym(A::AbstractMatrix)
     m, n = size(A)
     m==n || return false

doc/stdlib/linalg.rst

@@ -17,7 +17,7 @@ Linear algebra functions in Julia are largely implemented by calling functions f
 .. function:: \\(A, B)
    :noindex:
 
-   Matrix division using a polyalgorithm. For input matrices ``A`` and ``B``, the result ``X`` is such that ``A*X == B`` when ``A`` is square.  The solver that is used depends upon the structure of ``A``.  A direct solver is used for upper- or lower triangular ``A``.  For Hermitian ``A`` (equivalent to symmetric ``A`` for non-complex ``A``) the BunchKaufman factorization is used.  Otherwise an LU factorization is used. For rectangular ``A`` the result is the minimum-norm least squares solution computed by reducing ``A`` to bidiagonal form and solving the bidiagonal least squares problem.  For sparse, square ``A`` the LU factorization (from UMFPACK) is used.
+   Matrix division using a polyalgorithm. For input matrices ``A`` and ``B``, the result ``X`` is such that ``A*X == B`` when ``A`` is square.  The solver that is used depends upon the structure of ``A``.  A direct solver is used for upper- or lower triangular ``A``.  For Hermitian ``A`` (equivalent to symmetric ``A`` for non-complex ``A``) the ``BunchKaufman`` factorization is used.  Otherwise an LU factorization is used. For rectangular ``A`` the result is the minimum-norm least squares solution computed by reducing ``A`` to bidiagonal form and solving the bidiagonal least squares problem.  For sparse, square ``A`` the LU factorization (from UMFPACK) is used.
 
 .. function:: dot(x, y)
 
@@ -272,7 +272,7 @@ Linear algebra functions in Julia are largely implemented by calling functions f
 
    Compute the ``p``-norm of a vector or the operator norm of a matrix ``A``, defaulting to the ``p=2``-norm.
 
-   For vectors, ``p`` can assume any numeric value (even though not all values produce a mathematically valid vector norm). In particular, `norm(A, Inf)`` returns the largest value in ``abs(A)``, whereas ``norm(A, -Inf)`` returns the smallest.
+   For vectors, ``p`` can assume any numeric value (even though not all values produce a mathematically valid vector norm). In particular, ``norm(A, Inf)`` returns the largest value in ``abs(A)``, whereas ``norm(A, -Inf)`` returns the smallest.
 
    For matrices, valid values of ``p`` are ``1``, ``2``, or ``Inf``. Use :func:`normfro` to compute the Frobenius norm.
 
@@ -282,7 +282,17 @@ Linear algebra functions in Julia are largely implemented by calling functions f
 
 .. function:: cond(M, [p])
 
-   Matrix condition number, computed using the ``p``-norm. Valid values for ``p`` are ``1``, ``2`` (default), or ``Inf``.
+   Condition number of the matrix ``M``, computed using the operator ``p``-norm. Valid values for ``p`` are ``1``, ``2`` (default), or ``Inf``.
+
+.. function:: condskeel(M, [x, p])
+
+   .. math::
+      \kappa_S(M, p) & = & \left\Vert \left\vert M \right\vert \left\vert M^{-1} \right\vert  \right\Vert_p \\
+      \kappa_S(M, x, p) & = & \left\Vert \left\vert M \right\vert \left\vert M^{-1} \right\vert \left\vert x \right\vert \right\Vert_p
+
+   Skeel condition number :math:`\kappa_S` of the matrix ``M``, optionally with respect to the vector ``x``, as computed using the operator ``p``-norm. ``p`` is ``Inf`` by default, if not provided. Valid values for ``p`` are ``1``, ``2``, or ``Inf``.
+
+   This quantity is also known in the literature as the Bauer condition number, relative condition number, or componentwise relative condition number.
 
 .. function:: trace(M)
 

test/linalg.jl

@@ -191,13 +191,39 @@ debug && println("Solve square general system of equations")
     x = a \ b
     @test_approx_eq_eps a*x b 80ε
 
-debug && println("Solve upper trianguler system")
+debug && println("Solve upper triangular system")
     x = triu(a) \ b
-    @test_approx_eq_eps triu(a)*x b 20000ε
+
+    #Test forward error [JIN 5705] if this is not a BigFloat
+    γ = n*ε/(1-n*ε)
+    if eltya != BigFloat
+        bigA = big(triu(a))
+        ̂x = bigA \ b
+        for i=1:size(b, 2)
+            @test norm(̂x[:,i]-x[:,i], Inf)/norm(x[:,i], Inf) <= abs(condskeel(bigA, x[:,i])*γ/(1-condskeel(bigA)*γ))
+        end
+    end
+    #Test backward error [JIN 5705]
+    for i=1:size(b, 2)
+        @test norm(abs(b[:,i] - triu(a)*x[:,i]), Inf) <= γ * norm(triu(a), Inf) * norm(x[:,i], Inf)
+    end
 
 debug && println("Solve lower triangular system")
     x = tril(a)\b
-    @test_approx_eq_eps tril(a)*x b 10000ε
+
+    #Test forward error [JIN 5705] if this is not a BigFloat
+    γ = n*ε/(1-n*ε)
+    if eltya != BigFloat
+        bigA = big(tril(a))
+        ̂x = bigA \ b
+        for i=1:size(b, 2)
+            @test norm(̂x[:,i]-x[:,i], Inf)/norm(x[:,i], Inf) <= abs(condskeel(bigA, x[:,i])*γ/(1-condskeel(bigA)*γ))
+        end
+    end
+    #Test backward error [JIN 5705]
+    for i=1:size(b, 2)
+        @test norm(abs(b[:,i] - tril(a)*x[:,i]), Inf) <= γ * norm(tril(a), Inf) * norm(x[:,i], Inf)
+    end
 
 debug && println("Test null")
     if eltya != BigFloat && eltyb != BigFloat # Revisit when implemented in julia